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The 1/N Expansion in Colored Tensor Models

R˘ azvan Gur˘ au Laboratoire d’Informatique de Paris-Nord, 2011

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

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

Matrix Models

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

Matrix Models

A success story: Matrix Models in two dimensions

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory.

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N.

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces.

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces. ◮ They undergo a phase transition (“condensation”) to a continuum theory of

large surfaces.

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces. ◮ They undergo a phase transition (“condensation”) to a continuum theory of

large surfaces. Physics:

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces. ◮ They undergo a phase transition (“condensation”) to a continuum theory of

large surfaces. Physics: critical phenomena,

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces. ◮ They undergo a phase transition (“condensation”) to a continuum theory of

large surfaces. Physics: critical phenomena, conformal field theory,

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces. ◮ They undergo a phase transition (“condensation”) to a continuum theory of

large surfaces. Physics: critical phenomena, conformal field theory, the theory of strong interactions,

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces. ◮ They undergo a phase transition (“condensation”) to a continuum theory of

large surfaces. Physics: critical phenomena, conformal field theory, the theory of strong interactions, string theory,

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces. ◮ They undergo a phase transition (“condensation”) to a continuum theory of

large surfaces. Physics: critical phenomena, conformal field theory, the theory of strong interactions, string theory, quantum gravity in D = 2, etc.

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces. ◮ They undergo a phase transition (“condensation”) to a continuum theory of

large surfaces. Physics: critical phenomena, conformal field theory, the theory of strong interactions, string theory, quantum gravity in D = 2, etc. Mathematics:

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces. ◮ They undergo a phase transition (“condensation”) to a continuum theory of

large surfaces. Physics: critical phenomena, conformal field theory, the theory of strong interactions, string theory, quantum gravity in D = 2, etc. Mathematics: knot theory,

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces. ◮ They undergo a phase transition (“condensation”) to a continuum theory of

large surfaces. Physics: critical phenomena, conformal field theory, the theory of strong interactions, string theory, quantum gravity in D = 2, etc. Mathematics: knot theory, number theory and the Riemann hypothesis,

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces. ◮ They undergo a phase transition (“condensation”) to a continuum theory of

large surfaces. Physics: critical phenomena, conformal field theory, the theory of strong interactions, string theory, quantum gravity in D = 2, etc. Mathematics: knot theory, number theory and the Riemann hypothesis, invariants

• f algebraic curves,

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces. ◮ They undergo a phase transition (“condensation”) to a continuum theory of

large surfaces. Physics: critical phenomena, conformal field theory, the theory of strong interactions, string theory, quantum gravity in D = 2, etc. Mathematics: knot theory, number theory and the Riemann hypothesis, invariants

• f algebraic curves, enumeration problems, etc.

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

Matrix Models

A success story: Matrix Models in two dimensions

◮ An ab initio combinatorial statistical theory. ◮ Have built in scales N. ◮ Generate ribbon graphs ↔ discretized surfaces. ◮ They undergo a phase transition (“condensation”) to a continuum theory of

large surfaces. Physics: critical phenomena, conformal field theory, the theory of strong interactions, string theory, quantum gravity in D = 2, etc. Mathematics: knot theory, number theory and the Riemann hypothesis, invariants

• f algebraic curves, enumeration problems, etc.

All these applications rely crucially on the “1/N” expansion!

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

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. Z(Q) =

• [dφ] e

−N

• 1

2

φa1a2δa1b1δa2b2φ∗

b1b2+λ φa1a2φa2a3φa3a1

• 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. Z(Q) =

• [dφ] e

−N

• 1

2

φa1a2δa1b1δa2b2φ∗

b1b2+λ φa1a2φa2a3φa3a1

• The vertex is a ribbon vertex because the field φ has two arguments.

1

φ

3 1

φ

2 3

φ

2 a a 1 2 3 a a a a a a a 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. Z(Q) =

• [dφ] e

−N

• 1

2

φa1a2δa1b1δa2b2φ∗

b1b2+λ φa1a2φa2a3φa3a1

• The vertex is a ribbon vertex because the field φ has two arguments. The lines

conserve the two arguments (thus having two strands).

1

φ

3 1

φ

2 3

φ

2 a a 1 2 3 a a a a a a a

1 2 2 1

a a b b 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. Z(Q) =

• [dφ] e

−N

• 1

2

φa1a2δa1b1δa2b2φ∗

b1b2+λ φa1a2φa2a3φa3a1

• 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.

1

φ

3 1

φ

2 3

φ

2 a a 1 2 3 a a a a a a a

1 2 2 1

a a b b 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. Z(Q) =

• [dφ] e

−N

• 1

2

φa1a2δa1b1δa2b2φ∗

b1b2+λ φa1a2φa2a3φa3a1

• 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.

1

φ

3 1

φ

2 3

φ

2 a a 1 2 3 a a a a a a a

1 2 2 1

a a b b

Z(Q) is a sum over ribbon Feynman graphs.

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

Amplitude of Ribbon Graphs

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

Amplitude of Ribbon Graphs

The Amplitude of a graph with N vertices is A = λN N−L+N

lines

δa1b1δa2b2

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

Amplitude of Ribbon Graphs

The Amplitude of a graph with N vertices is A = λN N−L+N

lines

δa1b1δa2b2

1 1 1 3 2 2 3

a a b a b b w

• δa1b1

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

Amplitude of Ribbon Graphs

The Amplitude of a graph with N vertices is A = λN N−L+N

lines

δa1b1δa2b2

3 1 1 1 3 2 2

a a a b b b w

• δa1b1δb1c1

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

Amplitude of Ribbon Graphs

The Amplitude of a graph with N vertices is A = λN N−L+N

lines

δa1b1δa2b2

3 2 2 3 1 1 1

a a a b b b w

• δa1b1δb1c1 . . . δw1a1

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

Amplitude of Ribbon Graphs

The Amplitude of a graph with N vertices is A = λN N−L+N

lines

δa1b1δa2b2

3 2 2 3 1 1 1

a a a b b b w

• δa1b1δb1c1 . . . δw1a1 =
• δa1a1 = N

A = λN NN −L+F = λN N2−2g(G) with gG is the genus of the graph.

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

Amplitude of Ribbon Graphs

The Amplitude of a graph with N vertices is A = λN N−L+N

lines

δa1b1δa2b2

3 2 2 3 1 1 1

a a a b b b w

• δa1b1δb1c1 . . . δw1a1 =
• δa1a1 = N

A = λN NN −L+F = λN N2−2g(G) with gG is the genus of the graph. 1/N expansion in the genus.

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

Amplitude of Ribbon Graphs

The Amplitude of a graph with N vertices is A = λN N−L+N

lines

δa1b1δa2b2

3 2 2 3 1 1 1

a a a b b b w

• δa1b1δb1c1 . . . δw1a1 =
• δa1a1 = N

A = λN NN −L+F = λN N2−2g(G) with gG is the genus of the graph. 1/N expansion in the genus. Planar graphs (gG = 0) dominate in the large N limit.

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 are Dual to Discrete Surfaces

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 are Dual to Discrete Surfaces

3 2 2 3 1 1 1

a a a b b b w

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 are Dual to Discrete Surfaces

3 2 2 3 1 1 1

a a a b b b w

Place a point in the middle of each face.

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 are Dual to Discrete Surfaces

3 2 2 3 1 1 1

a a a b b b w

Place a point in the middle of each face. Draw a line crossing each ribbon line.

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 are Dual to Discrete Surfaces

3 2 2 3 1 1 1

a a a b b b w

Place a point in the middle of each face. Draw a line crossing each ribbon line. The ribbon vertices correspond to triangles.

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 are Dual to Discrete Surfaces

3 2 2 3 1 1 1

a a a b b b w

Place a point in the middle of each face. Draw a line crossing each ribbon line. The ribbon vertices correspond to triangles. A ribbon graph encodes unambiguously a gluing of triangles.

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 are Dual to Discrete Surfaces

3 2 2 3 1 1 1

a a a b b b w

Place a point in the middle of each face. Draw a line crossing each ribbon line. The ribbon vertices correspond to triangles. A ribbon graph encodes unambiguously a gluing of triangles. Matrix models sum over all graphs (i.e. surfaces) with canonical weights (Feynman rules).

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 are Dual to Discrete Surfaces

3 2 2 3 1 1 1

a a a b b b w

Place a point in the middle of each face. Draw a line crossing each ribbon line. The ribbon vertices correspond to triangles. 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

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

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

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 D dimensional spaces ↔ colored stranded graphs

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

From Matrix to COLORED Tensor Models

surfaces ↔ ribbon graphs D dimensional spaces ↔ colored stranded graphs Matrix Mab, S = N

• Mab ¯

Mab + λMabMbcMca

• 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

From Matrix to COLORED Tensor Models

surfaces ↔ ribbon graphs D dimensional spaces ↔ colored stranded graphs Matrix Mab, S = N

• Mab ¯

Mab + λMabMbcMca

• Tensors T i a1...aD with color i

S = ND/2 T i

... ¯

T i

... + λT 0 ...T 1 ... . . . T D ...

• 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

From Matrix to COLORED Tensor Models

surfaces ↔ ribbon graphs D dimensional spaces ↔ colored stranded graphs Matrix Mab, S = N

• Mab ¯

Mab + λMabMbcMca

• Tensors T i a1...aD with color i

S = ND/2 T i

... ¯

T i

... + λT 0 ...T 1 ... . . . T D ...

• g(G) ≥ 0 genus

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

From Matrix to COLORED Tensor Models

surfaces ↔ ribbon graphs D dimensional spaces ↔ colored stranded graphs Matrix Mab, S = N

• Mab ¯

Mab + λMabMbcMca

• Tensors T i a1...aD with color i

S = ND/2 T i

... ¯

T i

... + λT 0 ...T 1 ... . . . T D ...

• g(G) ≥ 0 genus

ω(G) ≥ 0 degree

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

From Matrix to COLORED Tensor Models

surfaces ↔ ribbon graphs D dimensional spaces ↔ colored stranded graphs Matrix Mab, S = N

• Mab ¯

Mab + λMabMbcMca

• Tensors T i a1...aD with color i

S = ND/2 T i

... ¯

T i

... + λT 0 ...T 1 ... . . . T D ...

• g(G) ≥ 0 genus

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

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

From Matrix to COLORED Tensor Models

surfaces ↔ ribbon graphs D dimensional spaces ↔ colored stranded graphs Matrix Mab, S = N

• Mab ¯

Mab + λMabMbcMca

• Tensors T i a1...aD with color i

S = ND/2 T i

... ¯

T i

... + λT 0 ...T 1 ... . . . T D ...

• g(G) ≥ 0 genus

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

2 (D−1)! ω(G) 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

From Matrix to COLORED Tensor Models

surfaces ↔ ribbon graphs D dimensional spaces ↔ colored stranded graphs Matrix Mab, S = N

• Mab ¯

Mab + λMabMbcMca

• Tensors T i a1...aD with color i

S = ND/2 T i

... ¯

T i

... + λT 0 ...T 1 ... . . . T D ...

• g(G) ≥ 0 genus

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

2 (D−1)! ω(G)

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

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

From Matrix to COLORED Tensor Models

surfaces ↔ ribbon graphs D dimensional spaces ↔ colored stranded graphs Matrix Mab, S = N

• Mab ¯

Mab + λMabMbcMca

• Tensors T i a1...aD with color i

S = ND/2 T i

... ¯

T i

... + λT 0 ...T 1 ... . . . T D ...

• g(G) ≥ 0 genus

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

2 (D−1)! ω(G)

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

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

Introduction Colored Tensor Models Colored Graphs Jackets and the 1/N expansion Topology Leading order graphs are spheres Conclusion

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

Colored Stranded Graphs

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

Colored Stranded Graphs

Clockwise and anticlockwise turning colored vertices (positive and negative oriented D simplices).

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

Colored Stranded Graphs

Clockwise and anticlockwise turning colored vertices (positive and negative oriented D simplices).

1

(0,1)

1

(0,1)

1 2 3 1

(0,1) (0,1)

2 4 3

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

Colored Stranded Graphs

Clockwise and anticlockwise turning colored vertices (positive and negative oriented D simplices).

1

(0,1)

1

(0,1)

1 2 3 1

(0,1) (0,1)

2 4 3

Lines have a well defined color and D parallel strands (D − 1 simplices).

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

Colored Stranded Graphs

Clockwise and anticlockwise turning colored vertices (positive and negative oriented D simplices).

1

(0,1)

1

(0,1)

1 2 3 1

(0,1) (0,1)

2 4 3

Lines have a well defined color and D parallel strands (D − 1 simplices).

1 1 1 1 1 1 1 1

(0,1) (0,1) (0,1) (0,1)

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

Colored Stranded Graphs

Clockwise and anticlockwise turning colored vertices (positive and negative oriented D simplices).

1

(0,1)

1

(0,1)

1 2 3 1

(0,1) (0,1)

2 4 3

Lines have a well defined color and D parallel strands (D − 1 simplices).

1 1 1 1 1 1 1 1

(0,1) (0,1) (0,1) (0,1)

Strands are identified by a couple of colors (D − 2 simplices).

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

Action

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

Action

Let T i

a1...aD, ¯

T i

a1...aD tensor fields with color i = 0 . . . D .

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

Action

Let T i

a1...aD, ¯

T i

a1...aD tensor fields with color i = 0 . . . D .

S = ND/2

i

¯ T i

a1...aDT i a1...aD + λ

• i

T i

aii−1...ai0aiD...aii+1 + ¯

λ

• i

¯ T i

aii−1...ai0aiD...aii+1

• 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

Action

Let T i

a1...aD, ¯

T i

a1...aD tensor fields with color i = 0 . . . D .

S = ND/2

i

¯ T i

a1...aDT i a1...aD + λ

• i

T i

aii−1...ai0aiD...aii+1 + ¯

λ

• i

¯ T i

aii−1...ai0aiD...aii+1

• Topology of the Colored Graphs

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

Action

Let T i

a1...aD, ¯

T i

a1...aD tensor fields with color i = 0 . . . D .

S = ND/2

i

¯ T i

a1...aDT i a1...aD + λ

• i

T i

aii−1...ai0aiD...aii+1 + ¯

λ

• i

¯ T i

aii−1...ai0aiD...aii+1

• Topology of the Colored Graphs

Amplitude of the graphs:

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

Action

Let T i

a1...aD, ¯

T i

a1...aD tensor fields with color i = 0 . . . D .

S = ND/2

i

¯ T i

a1...aDT i a1...aD + λ

• i

T i

aii−1...ai0aiD...aii+1 + ¯

λ

• i

¯ T i

aii−1...ai0aiD...aii+1

• Topology of the Colored Graphs

Amplitude of the graphs:

◮ the N = 2p vertices of a graph bring each ND/2

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

Action

Let T i

a1...aD, ¯

T i

a1...aD tensor fields with color i = 0 . . . D .

S = ND/2

i

¯ T i

a1...aDT i a1...aD + λ

• i

T i

aii−1...ai0aiD...aii+1 + ¯

λ

• i

¯ T i

aii−1...ai0aiD...aii+1

• Topology of the Colored Graphs

Amplitude of the graphs:

◮ the N = 2p vertices of a graph bring each ND/2 ◮ the L lines of a graphs bring each N−D/2

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

Action

Let T i

a1...aD, ¯

T i

a1...aD tensor fields with color i = 0 . . . D .

S = ND/2

i

¯ T i

a1...aDT i a1...aD + λ

• i

T i

aii−1...ai0aiD...aii+1 + ¯

λ

• i

¯ T i

aii−1...ai0aiD...aii+1

• Topology of the Colored Graphs

Amplitude of the graphs:

◮ the N = 2p vertices of a graph bring each ND/2 ◮ the L lines of a graphs bring each N−D/2 ◮ the F faces of a graph bring each N

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

Action

Let T i

a1...aD, ¯

T i

a1...aD tensor fields with color i = 0 . . . D .

S = ND/2

i

¯ T i

a1...aDT i a1...aD + λ

• i

T i

aii−1...ai0aiD...aii+1 + ¯

λ

• i

¯ T i

aii−1...ai0aiD...aii+1

• Topology of the Colored Graphs

Amplitude of the graphs:

◮ the N = 2p vertices of a graph bring each ND/2 ◮ the L lines of a graphs bring each N−D/2 ◮ the F faces of a graph bring each N

AG = (λ¯ λ)p N−L D

2 +N D 2 +F 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

Action

Let T i

a1...aD, ¯

T i

a1...aD tensor fields with color i = 0 . . . D .

S = ND/2

i

¯ T i

a1...aDT i a1...aD + λ

• i

T i

aii−1...ai0aiD...aii+1 + ¯

λ

• i

¯ T i

aii−1...ai0aiD...aii+1

• Topology of the Colored Graphs

Amplitude of the graphs:

◮ the N = 2p vertices of a graph bring each ND/2 ◮ the L lines of a graphs bring each N−D/2 ◮ the F faces of a graph bring each N

AG = (λ¯ λ)p N−L D

2 +N D 2 +F

But N(D + 1) = 2L ⇒ L = (D + 1)p

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

Action

Let T i

a1...aD, ¯

T i

a1...aD tensor fields with color i = 0 . . . D .

S = ND/2

i

¯ T i

a1...aDT i a1...aD + λ

• i

T i

aii−1...ai0aiD...aii+1 + ¯

λ

• i

¯ T i

aii−1...ai0aiD...aii+1

• Topology of the Colored Graphs

Amplitude of the graphs:

◮ the N = 2p vertices of a graph bring each ND/2 ◮ the L lines of a graphs bring each N−D/2 ◮ the F faces of a graph bring each N

AG = (λ¯ λ)p N−L D

2 +N D 2 +F = (λ¯

λ)p N−p D(D−1)

2

+F

But N(D + 1) = 2L ⇒ L = (D + 1)p

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

Action

Let T i

a1...aD, ¯

T i

a1...aD tensor fields with color i = 0 . . . D .

S = ND/2

i

¯ T i

a1...aDT i a1...aD + λ

• i

T i

aii−1...ai0aiD...aii+1 + ¯

λ

• i

¯ T i

aii−1...ai0aiD...aii+1

• Topology of the Colored Graphs

Amplitude of the graphs:

◮ the N = 2p vertices of a graph bring each ND/2 ◮ the L lines of a graphs bring each N−D/2 ◮ the F faces of a graph bring each N

AG = (λ¯ λ)p N−L D

2 +N D 2 +F = (λ¯

λ)p N−p D(D−1)

2

+F

But N(D + 1) = 2L ⇒ L = (D + 1)p

Compute F !

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

Introduction Colored Tensor Models Colored Graphs Jackets and the 1/N expansion Topology Leading order graphs are spheres Conclusion

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

Jackets 1

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

Jackets 1

Define simpler graphs.

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

1 2 2 3 1 3

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

1 2 2 3 1 3

02 and 13: opposing edges of the tetrahedron.

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

1 2 2 3 1 3

02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent.

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

1 2 2 3 1 3

02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) graphs.

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

1 2 2 3 1 3

02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) graphs.

1 2 1 2 3 3 1 2 3

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

1 2 2 3 1 3

02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) graphs.

1 2 1 2 3 3 1 2 3

1 2

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

1 2 2 3 1 3

02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) graphs.

1 2 1 2 3 3 1 2 3

1 2

0, 1, 2, . . .

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

1 2 2 3 1 3

02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) graphs.

1 2 1 2 3 3 1 2 3

1 2

0, 1, 2, . . .

π(0) π (0)

2

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

1 2 2 3 1 3

02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) graphs.

1 2 1 2 3 3 1 2 3

1 2

0, 1, 2, . . .

π(0) π (0)

2

0, π(0), π2(0), . . .

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

1 2 2 3 1 3

02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) graphs.

1 2 1 2 3 3 1 2 3

1 2

0, 1, 2, . . .

π(0) π (0)

2

0, π(0), π2(0), . . .

1 2D! jackets.

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

1 2 2 3 1 3

02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) graphs.

1 2 1 2 3 3 1 2 3

1 2

0, 1, 2, . . .

π(0) π (0)

2

0, π(0), π2(0), . . .

1 2D! jackets. Contain all the vertices and

all the lines of G.

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

1 2 2 3 1 3

02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) graphs.

1 2 1 2 3 3 1 2 3

1 2

0, 1, 2, . . .

π(0) π (0)

2

0, π(0), π2(0), . . .

1 2D! jackets. Contain all the vertices and

all the lines of G. A face belongs to (D − 1)! jackets.

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

Jackets 1

Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph.

1 2 2 3 1 3

02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) graphs.

1 2 1 2 3 3 1 2 3

1 2

0, 1, 2, . . .

π(0) π (0)

2

0, π(0), π2(0), . . .

1 2D! jackets. Contain all the vertices and

all the lines of G. A face belongs to (D − 1)! jackets. The degree of G is ω(G) =

J gJ .

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

Jackets 2: Jackets and Amplitude

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

Jackets 2: Jackets and Amplitude

Theorem F and ω(G) are related by F = 1 2D(D − 1)p + D − 2 (D − 1)!ω(G)

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

Jackets 2: Jackets and Amplitude

Theorem F and ω(G) are related by F = 1 2D(D − 1)p + D − 2 (D − 1)!ω(G) Proof: N = 2p, L = (D + 1)p

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

Jackets 2: Jackets and Amplitude

Theorem F and ω(G) are related by F = 1 2D(D − 1)p + D − 2 (D − 1)!ω(G) Proof: N = 2p, L = (D + 1)p For each jacket J , 2p − (D + 1)p + FJ = 2 − 2gJ .

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

Jackets 2: Jackets and Amplitude

Theorem F and ω(G) are related by F = 1 2D(D − 1)p + D − 2 (D − 1)!ω(G) Proof: N = 2p, L = (D + 1)p For each jacket J , 2p − (D + 1)p + FJ = 2 − 2gJ . Sum over the jackets: (D − 1)!F =

J FJ = 1 2D!(D − 1)p + D! − 2 J gJ

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

Jackets 2: Jackets and Amplitude

Theorem F and ω(G) are related by F = 1 2D(D − 1)p + D − 2 (D − 1)!ω(G) Proof: N = 2p, L = (D + 1)p For each jacket J , 2p − (D + 1)p + FJ = 2 − 2gJ . Sum over the jackets: (D − 1)!F =

J FJ = 1 2D!(D − 1)p + D! − 2 J gJ

The amplitude of a graph is given by its degree AG = (λ¯ λ)p N−p D(D−1)

2

+F = (λ¯

λ)p ND−

2 (D−1)! ω(G) 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

Introduction Colored Tensor Models Colored Graphs Jackets and the 1/N expansion Topology Leading order graphs are spheres Conclusion

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

Topology 1: Colored vs. Stranded Graphs

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

Topology 1: Colored vs. Stranded Graphs

THEOREM: [M. Ferri and C. Gagliardi, ’82] Any D-dimensional piecewise linear

• rientable manifold admits a colored triangulation.

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

Topology 1: Colored vs. Stranded Graphs

THEOREM: [M. Ferri and C. Gagliardi, ’82] Any D-dimensional piecewise linear

• rientable manifold admits a colored triangulation.

We have clockwise and anticlockwise turning vertices.

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

Topology 1: Colored vs. Stranded Graphs

THEOREM: [M. Ferri and C. Gagliardi, ’82] Any D-dimensional piecewise linear

• rientable manifold admits a colored triangulation.

We have clockwise and anticlockwise turning vertices. Lines connect opposing vertices and have a color index.

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

Topology 1: Colored vs. Stranded Graphs

THEOREM: [M. Ferri and C. Gagliardi, ’82] Any D-dimensional piecewise linear

• rientable manifold admits a colored triangulation.

We have clockwise and anticlockwise turning vertices. Lines connect opposing vertices and have a color index. All the information is encoded in the colors

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

Topology 1: Colored vs. Stranded Graphs

THEOREM: [M. Ferri and C. Gagliardi, ’82] Any D-dimensional piecewise linear

• rientable manifold admits a colored triangulation.

We have clockwise and anticlockwise turning vertices. Lines connect opposing vertices and have a color index. All the information is encoded in the colors

1 2 3

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

Topology 1: Colored vs. Stranded Graphs

THEOREM: [M. Ferri and C. Gagliardi, ’82] Any D-dimensional piecewise linear

• rientable manifold admits a colored triangulation.

We have clockwise and anticlockwise turning vertices. Lines connect opposing vertices and have a color index. All the information is encoded in the colors

1 2 3

represented as

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

Topology 1: Colored vs. Stranded Graphs

THEOREM: [M. Ferri and C. Gagliardi, ’82] Any D-dimensional piecewise linear

• rientable manifold admits a colored triangulation.

We have clockwise and anticlockwise turning vertices. Lines connect opposing vertices and have a color index. All the information is encoded in the colors

1 2 3

represented as 3 2 1

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

Topology 1: Colored vs. Stranded Graphs

THEOREM: [M. Ferri and C. Gagliardi, ’82] Any D-dimensional piecewise linear

• rientable manifold admits a colored triangulation.

We have clockwise and anticlockwise turning vertices. Lines connect opposing vertices and have a color index. All the information is encoded in the colors

1 2 3

represented as 3 2 1 Conversely: expand the vertices into stranded vertices and the lines into stranded lines with parallel strands

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

Topology 1: Colored vs. Stranded Graphs

THEOREM: [M. Ferri and C. Gagliardi, ’82] Any D-dimensional piecewise linear

• rientable manifold admits a colored triangulation.

We have clockwise and anticlockwise turning vertices. Lines connect opposing vertices and have a color index. All the information is encoded in the colors

1 2 3

represented as 3 2 1 Conversely: expand the vertices into stranded vertices and the lines into stranded lines with parallel strands

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

Topology 2: Bubbles

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

Topology 2: Bubbles

The vertices of G are subgraphs with 0 colors.

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

Topology 2: Bubbles

The vertices of G are subgraphs with 0 colors. The lines are subgraphs with exactly 1 color.

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

Topology 2: Bubbles

The vertices of G are subgraphs with 0 colors. The lines are subgraphs with exactly 1 color. The faces are subgraphs with exactly 2 colors.

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

Topology 2: Bubbles

The vertices of G are subgraphs with 0 colors. The lines are subgraphs with exactly 1 color. The faces are subgraphs with exactly 2 colors. The n-bubbles are the maximally connected subgraphs with n fixed colors (denoted Bi1...in

(σ) , with i1 < · · · < in the colors).

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

Topology 2: Bubbles

The vertices of G are subgraphs with 0 colors. The lines are subgraphs with exactly 1 color. The faces are subgraphs with exactly 2 colors. The n-bubbles are the maximally connected subgraphs with n fixed colors (denoted Bi1...in

(σ) , with i1 < · · · < in the colors). 1 1 2 3 1 1 1 2 2 2 3 3 3 2 1 3 2 3 3 1 2 3 1 2

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

Topology 2: Bubbles

The vertices of G are subgraphs with 0 colors. The lines are subgraphs with exactly 1 color. The faces are subgraphs with exactly 2 colors. The n-bubbles are the maximally connected subgraphs with n fixed colors (denoted Bi1...in

(σ) , with i1 < · · · < in the colors). 1 1 2 3 1 1 1 2 2 2 3 3 3 2 1 3 2 3 3 1 2 3 1 2

A colored graph G is dual to an orientable, normal, D dimensional, simplicial pseudo manifold. Its n-bubbles are dual to the links of the D − n simplices of the pseudo manifold.

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

Topology 3: Homeomorphisms and 1-Dipoles

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

Topology 3: Homeomorphisms and 1-Dipoles

D D D v w 1 2 1 2 1 2

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

Topology 3: Homeomorphisms and 1-Dipoles

D D D v w 1 2 1 2 1 2

A 1-dipole: a line (say of color 0) connecting two vertices v ∈ B1...D

(α)

and w ∈ B1...D

(β)

with B1...D

(α)

= B1...D

(β) .

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

Topology 3: Homeomorphisms and 1-Dipoles

D D D v w 1 2 1 2 1 2

A 1-dipole: a line (say of color 0) connecting two vertices v ∈ B1...D

(α)

and w ∈ B1...D

(β)

with B1...D

(α)

= B1...D

(β) .

A 1-Dipole can be contracted, that is the lines together with the vertices v and w can be deleted from G and the remaining lines reconnected respecting the coloring. Call the graph after contraction G/d.

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

Topology 3: Homeomorphisms and 1-Dipoles

D D D v w 1 2 1 2 1 2

A 1-dipole: a line (say of color 0) connecting two vertices v ∈ B1...D

(α)

and w ∈ B1...D

(β)

with B1...D

(α)

= B1...D

(β) .

A 1-Dipole can be contracted, that is the lines together with the vertices v and w can be deleted from G and the remaining lines reconnected respecting the coloring. Call the graph after contraction G/d. THEOREM: [M. Ferri and C. Gagliardi, ’82] If either B1...D

(α)

• r B1...D

(β)

is dual to a sphere, then the two pseudo manifolds dual to G and G/d are homeomorphic.

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

Topology 3: Homeomorphisms and 1-Dipoles

D D D v w 1 2 1 2 1 2

A 1-dipole: a line (say of color 0) connecting two vertices v ∈ B1...D

(α)

and w ∈ B1...D

(β)

with B1...D

(α)

= B1...D

(β) .

A 1-Dipole can be contracted, that is the lines together with the vertices v and w can be deleted from G and the remaining lines reconnected respecting the coloring. Call the graph after contraction G/d. THEOREM: [M. Ferri and C. Gagliardi, ’82] If either B1...D

(α)

• r B1...D

(β)

is dual to a sphere, then the two pseudo manifolds dual to G and G/d are homeomorphic. It is in principle very difficult to check if a bubble is a sphere or not.

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

Introduction Colored Tensor Models Colored Graphs Jackets and the 1/N expansion Topology Leading order graphs are spheres Conclusion

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

Jackets, Bubbles, 1-Dipoles

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

Jackets, Bubbles, 1-Dipoles

The D-bubbles B

i (ρ) of G are graphs with D colors, thus they admit jackets and

have a degree.

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

Jackets, Bubbles, 1-Dipoles

The D-bubbles B

i (ρ) of G are graphs with D colors, thus they admit jackets and

have a degree. The degrees of G and of its bubbles are not independent.

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

Jackets, Bubbles, 1-Dipoles

The D-bubbles B

i (ρ) of G are graphs with D colors, thus they admit jackets and

have a degree. The degrees of G and of its bubbles are not independent. Theorem ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

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

Jackets, Bubbles, 1-Dipoles

The D-bubbles B

i (ρ) of G are graphs with D colors, thus they admit jackets and

have a degree. The degrees of G and of its bubbles are not independent. Theorem ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

Theorem The degree of the graph is invariant under 1-Dipole moves, ω(G) = ω(G/d)

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

Degree 0 Graphs are Spheres

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

Degree 0 Graphs are Spheres

ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

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

Degree 0 Graphs are Spheres

ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

In a graph G with 2p vertices and B[D] D-bubbles I contract a full set of 1-Dipoles and bring it to Gf with 2pf vertices and exactly one D-bubble for each colors i.

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

Degree 0 Graphs are Spheres

ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

In a graph G with 2p vertices and B[D] D-bubbles I contract a full set of 1-Dipoles and bring it to Gf with 2pf vertices and exactly one D-bubble for each colors i. Every contraction: p → p − 1, B[D] → B[D] − 1

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

Degree 0 Graphs are Spheres

ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

In a graph G with 2p vertices and B[D] D-bubbles I contract a full set of 1-Dipoles and bring it to Gf with 2pf vertices and exactly one D-bubble for each colors i. Every contraction: p → p − 1, B[D] → B[D] − 1 p − pf = B[D] − B[D]

f

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

Degree 0 Graphs are Spheres

ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

In a graph G with 2p vertices and B[D] D-bubbles I contract a full set of 1-Dipoles and bring it to Gf with 2pf vertices and exactly one D-bubble for each colors i. Every contraction: p → p − 1, B[D] → B[D] − 1 p − pf = B[D] − B[D]

f

= B[D] − (D + 1) ⇒ p + D − B[D] = pf − 1 ≥ 0

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

Degree 0 Graphs are Spheres

ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

In a graph G with 2p vertices and B[D] D-bubbles I contract a full set of 1-Dipoles and bring it to Gf with 2pf vertices and exactly one D-bubble for each colors i. Every contraction: p → p − 1, B[D] → B[D] − 1 p − pf = B[D] − B[D]

f

= B[D] − (D + 1) ⇒ p + D − B[D] = pf − 1 ≥ 0 Thus ω(G) = 0 ⇒ ω(B

i (ρ)) = 0.

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

Degree 0 Graphs are Spheres

ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

In a graph G with 2p vertices and B[D] D-bubbles I contract a full set of 1-Dipoles and bring it to Gf with 2pf vertices and exactly one D-bubble for each colors i. Every contraction: p → p − 1, B[D] → B[D] − 1 p − pf = B[D] − B[D]

f

= B[D] − (D + 1) ⇒ p + D − B[D] = pf − 1 ≥ 0 Thus ω(G) = 0 ⇒ ω(B

i (ρ)) = 0.

Theorem If ω(G) = 0 then G is dual to a D-dimensional sphere.

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

Degree 0 Graphs are Spheres

ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

In a graph G with 2p vertices and B[D] D-bubbles I contract a full set of 1-Dipoles and bring it to Gf with 2pf vertices and exactly one D-bubble for each colors i. Every contraction: p → p − 1, B[D] → B[D] − 1 p − pf = B[D] − B[D]

f

= B[D] − (D + 1) ⇒ p + D − B[D] = pf − 1 ≥ 0 Thus ω(G) = 0 ⇒ ω(B

i (ρ)) = 0.

Theorem If ω(G) = 0 then G is dual to a D-dimensional sphere. Proof: Induction on D.

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

Degree 0 Graphs are Spheres

ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

In a graph G with 2p vertices and B[D] D-bubbles I contract a full set of 1-Dipoles and bring it to Gf with 2pf vertices and exactly one D-bubble for each colors i. Every contraction: p → p − 1, B[D] → B[D] − 1 p − pf = B[D] − B[D]

f

= B[D] − (D + 1) ⇒ p + D − B[D] = pf − 1 ≥ 0 Thus ω(G) = 0 ⇒ ω(B

i (ρ)) = 0.

Theorem If ω(G) = 0 then G is dual to a D-dimensional sphere. Proof: Induction on D. D = 2: the colored graphs are ribbon graphs and the degree is the genus.

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

Degree 0 Graphs are Spheres

ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

In a graph G with 2p vertices and B[D] D-bubbles I contract a full set of 1-Dipoles and bring it to Gf with 2pf vertices and exactly one D-bubble for each colors i. Every contraction: p → p − 1, B[D] → B[D] − 1 p − pf = B[D] − B[D]

f

= B[D] − (D + 1) ⇒ p + D − B[D] = pf − 1 ≥ 0 Thus ω(G) = 0 ⇒ ω(B

i (ρ)) = 0.

Theorem If ω(G) = 0 then G is dual to a D-dimensional sphere. Proof: Induction on D. D = 2: the colored graphs are ribbon graphs and the degree is the genus. In D > 2, ω(G) = 0 ⇒ ω(B

i (ρ)) = 0 and all ω(B i (ρ)) are a

spheres by the induction hypothesis.

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

Degree 0 Graphs are Spheres

ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

In a graph G with 2p vertices and B[D] D-bubbles I contract a full set of 1-Dipoles and bring it to Gf with 2pf vertices and exactly one D-bubble for each colors i. Every contraction: p → p − 1, B[D] → B[D] − 1 p − pf = B[D] − B[D]

f

= B[D] − (D + 1) ⇒ p + D − B[D] = pf − 1 ≥ 0 Thus ω(G) = 0 ⇒ ω(B

i (ρ)) = 0.

Theorem If ω(G) = 0 then G is dual to a D-dimensional sphere. Proof: Induction on D. D = 2: the colored graphs are ribbon graphs and the degree is the genus. In D > 2, ω(G) = 0 ⇒ ω(B

i (ρ)) = 0 and all ω(B i (ρ)) are a

spheres by the induction hypothesis. 1-Dipole contractions do not change the degree and are homeomorphisms. Gf is homeomorphic with G and has pf = 1.

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

Degree 0 Graphs are Spheres

ω(G) = (D−1)!

2

• p + D − B[D]

+

i,ρ ω(B i (ρ))

In a graph G with 2p vertices and B[D] D-bubbles I contract a full set of 1-Dipoles and bring it to Gf with 2pf vertices and exactly one D-bubble for each colors i. Every contraction: p → p − 1, B[D] → B[D] − 1 p − pf = B[D] − B[D]

f

= B[D] − (D + 1) ⇒ p + D − B[D] = pf − 1 ≥ 0 Thus ω(G) = 0 ⇒ ω(B

i (ρ)) = 0.

Theorem If ω(G) = 0 then G is dual to a D-dimensional sphere. Proof: Induction on D. D = 2: the colored graphs are ribbon graphs and the degree is the genus. In D > 2, ω(G) = 0 ⇒ ω(B

i (ρ)) = 0 and all ω(B i (ρ)) are a

spheres by the induction hypothesis. 1-Dipole contractions do not change the degree and are homeomorphisms. Gf is homeomorphic with G and has pf = 1. The only graph with pf = 1 is a sphere.

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

From Matrix to COLORED Tensor Models

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

From Matrix to COLORED Tensor Models

Tensors T i a1...aD with color i S = ND/2 T i

... ¯

T i

... + λT 0 ...T 1 ... . . . T D ... + ¯

λ ¯ T 0

... ¯

T 1

... . . . ¯

T D

...

• 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

From Matrix to COLORED Tensor Models

Tensors T i a1...aD with color i S = ND/2 T i

... ¯

T i

... + λT 0 ...T 1 ... . . . T D ... + ¯

λ ¯ T 0

... ¯

T 1

... . . . ¯

T D

...

• ω(G) =

J gJ ≥ 0 degree

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

From Matrix to COLORED Tensor Models

Tensors T i a1...aD with color i S = ND/2 T i

... ¯

T i

... + λT 0 ...T 1 ... . . . T D ... + ¯

λ ¯ T 0

... ¯

T 1

... . . . ¯

T D

...

• ω(G) =

J gJ ≥ 0 degree

1/N expansion in the degree A(G) = ND−

2 (D−1)! ω(G) 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

From Matrix to COLORED Tensor Models

Tensors T i a1...aD with color i S = ND/2 T i

... ¯

T i

... + λT 0 ...T 1 ... . . . T D ... + ¯

λ ¯ T 0

... ¯

T 1

... . . . ¯

T D

...

• ω(G) =

J gJ ≥ 0 degree

1/N expansion in the degree A(G) = ND−

2 (D−1)! ω(G)

colored stranded graphs ↔ D dimensional pseudo manifolds

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

From Matrix to COLORED Tensor Models

Tensors T i a1...aD with color i S = ND/2 T i

... ¯

T i

... + λT 0 ...T 1 ... . . . T D ... + ¯

λ ¯ T 0

... ¯

T 1

... . . . ¯

T D

...

• ω(G) =

J gJ ≥ 0 degree

1/N expansion in the degree A(G) = ND−

2 (D−1)! ω(G)

colored stranded graphs ↔ D dimensional pseudo manifolds leading order: ω(G) = 0 are spheres

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

Conclusion: A To Do List

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

Conclusion: A To Do List

◮ Is the dominant sector summable?

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

Conclusion: A To Do List

◮ Is the dominant sector summable? ◮ Does it lead to a phase transition and a continuum theory?

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

Conclusion: A To Do List

◮ Is the dominant sector summable? ◮ Does it lead to a phase transition and a continuum theory? ◮ What are the critical exponents?

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

Conclusion: A To Do List

◮ Is the dominant sector summable? ◮ Does it lead to a phase transition and a continuum theory? ◮ What are the critical exponents? ◮ Multi critical points?

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

Conclusion: A To Do List

◮ Is the dominant sector summable? ◮ Does it lead to a phase transition and a continuum theory? ◮ What are the critical exponents? ◮ Multi critical points? ◮ More complex models, driven to the phase transition by renormalization group

flow.

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

Conclusion: A To Do List

◮ Is the dominant sector summable? ◮ Does it lead to a phase transition and a continuum theory? ◮ What are the critical exponents? ◮ Multi critical points? ◮ More complex models, driven to the phase transition by renormalization group

flow.

◮ Generalize the results obtained using matrix models in higher dimensions.

22