Edge–colored graphs as higher-dimensional maps Valentin Bonzom LIPN, Paris University 13 October 17, 2016 RGP2016
Discretization of manifolds ◮ 2D discrete surfaces: triangulations, p –angulations and combinatorial maps ◮ 3D triangulations: gluings of tetrahedra
Discretization of manifolds ◮ 2D discrete surfaces: triangulations, p –angulations and combinatorial maps ◮ 3D triangulations: gluings of tetrahedra ◮ How to represent them in a suitable fashion for combinatorics? ◮ Equivalent of p –angulations? ◮ Enumeration? ◮ Topological recursion, integrable hierarchies?
Combinatorial maps Graph with cyclic ordering of edges incident to each vertex � =
Combinatorial maps Graph with cyclic ordering of edges incident to each vertex � = Cyclic ordering defines faces: follow the corners
2 p –angulation ◮ Faces of degree 2 p ◮ Duality: vertices of degree 2 p ◮ Euler’s relation with E ( M ) = pV ( M ) F ( M ) − E ( M ) + V ( M ) = F ( M ) − ( p − 1) V ( M ) = 2 − 2 g ( M ) Two properties of the genus to generalize ◮ g ( M ) ≥ 0 ⇒ bound on F ( M ) linear in V ( M ) ◮ Maximizing F ( M ) at fixed V ( M ) equivalent to g ( M ) = 0 Main challenge: generalize that!
2 p –angulation ◮ Faces of degree 2 p ◮ Duality: vertices of degree 2 p ◮ Euler’s relation with E ( M ) = pV ( M ) F ( M ) − E ( M ) + V ( M ) = F ( M ) − ( p − 1) V ( M ) = 2 − 2 g ( M ) Two properties of the genus to generalize ◮ g ( M ) ≥ 0 ⇒ bound on F ( M ) linear in V ( M ) ◮ Maximizing F ( M ) at fixed V ( M ) equivalent to g ( M ) = 0 Main challenge: generalize that!
2 p –angulation ◮ Faces of degree 2 p ◮ Duality: vertices of degree 2 p ◮ Euler’s relation with E ( M ) = pV ( M ) F ( M ) − E ( M ) + V ( M ) = F ( M ) − ( p − 1) V ( M ) = 2 − 2 g ( M ) Two properties of the genus to generalize ◮ g ( M ) ≥ 0 ⇒ bound on F ( M ) linear in V ( M ) ◮ Maximizing F ( M ) at fixed V ( M ) equivalent to g ( M ) = 0 Main challenge: generalize that!
What do we know? Maps: from Tutte to today Enumeration ◮ Count maps with possible decorations (Ising, Potts, loops) ◮ Exact generating functions or their properties ◮ Random matrix model techniques ◮ Bijections [Cori-Vauquelin-Schaeffer, Bouttier-Di Francesco-Guitter] ◮ Tutte’s equations and topological recursion [Borot, Eynard, Orantin] Physics motivations and applications ◮ Two-dimensional quantum gravity coupled to matter ◮ Liouville theory coupled to conformal field theories ◮ Statistical mechanics ◮ Celebrated KPZ relations
What do we know? II Geometric applications of bijections ◮ Two-point, three-point functions ◮ Local limit ◮ Continuum limit [Brownian sphere] Enumerative geometry ◮ Intersection numbers [Kontsevich-Witten] ◮ Hurwitz numbers ◮ Integrable hierarchies, etc.
In higher dimensions First approach based on generalizing random matrices � dM e − N (tr M 2 + t tr M 4 ) = � N 2 − 2 g ( Q ) t n ( Q ) ln H N × N quadrangulations Q ◮ From matrices to tensors Local rules T cde M a 1 a 2 − → T a 1 a 2 ... a d T abc [Ambjorn-Durhuus-Jonsson, Gross, Sasakura] all in 91 T abc T ebf ◮ Tensor integrals generate T def stranded graphs T fda
Problems with tensors and stranded graphs 3 origins of difficulty ◮ Random tensor techniques: non-existent Contrast with random matrices: eigenvalues and orthogonal polynomials e.g. ◮ So look at the stranded graphs directly?
Problems with tensors and stranded graphs 3 origins of difficulty ◮ Random tensor techniques: non-existent Contrast with random matrices: eigenvalues and orthogonal polynomials e.g. ◮ So look at the stranded graphs directly? ◮ Topology: One can associate topology to graph, very singular! ◮ Combinatorics ◮ Closed strands represent simplices of codimension 2 ex: vertices in 2D, edges in 3D, triangles in 4D, etc. ◮ One needs to count/classify graphs w.r.t. number of closed strands Nearly impossible! ◮ As a result, essential missing piece is generalization of genus! No analytical progress for 20 years!
Objectives Objectives: Family of triangulations s.t. ◮ No topological restriction ◮ Extension of genus: degree ω ( T ) of T ∆ d − 2 ( T ) − α ∆ d ( T ) = d − ω ( T ) , ω ( T ) ≥ 0 ◮ Infinite number of triangulations at fixed ω ◮ Balance on α Possible with colored triangulations!
How to represent triangulations? Triangulations ◮ Gluing of simplices (tetrahedra, pentachora, etc.) ◮ Defined by attaching maps ◮ Ensemble of triangulations defined by constraints on attaching maps Various ensembles ◮ Various ensembles in topology (simplicial, CW, ∆–complexes, etc.) ◮ Not suitable for combinatorics (too wild) ◮ Digging through old work, found colored triangulations Crystallization, graph–encoded manifold: Cristofori’s talk! ◮ Nice for combinatorics: represented by edge–colored graphs
Colored graphs 1 1 ( d + 1)-colored graphs 2 2 2 2 3 3 3 ◮ Bipartite graphs 3 black and white vertices 1 1 1 1 ◮ Edges colored with d + 1 2 2 possible colors 1 1 ◮ Vertices of degree d + 1 3 2 2 2 2 3 3 ◮ All colors incident exactly 3 3 1 1 1 1 once at each vertex 3 2 2
Colored graphs 1 ( d + 1)-colored graphs ◮ Bipartite graphs black and white vertices 3 2 2 ◮ Edges colored with d + 1 possible colors 3 ◮ Vertices of degree d + 1 1 1 3 ◮ All colors incident exactly once at each vertex 2 Faces are closed cycles with only two colors.
Colored graphs 1 ( d + 1)-colored graphs ◮ Bipartite graphs black and white vertices 3 2 2 ◮ Edges colored with d + 1 possible colors 3 ◮ Vertices of degree d + 1 1 1 3 ◮ All colors incident exactly once at each vertex 2 Faces are closed cycles with only two colors.
Colored graphs 1 ( d + 1)-colored graphs ◮ Bipartite graphs black and white vertices 3 2 2 ◮ Edges colored with d + 1 possible colors 3 ◮ Vertices of degree d + 1 1 1 3 ◮ All colors incident exactly once at each vertex 2 Faces are closed cycles with only two colors.
Triangulations from colored graphs vertex d –simplex → edge ( d − 1)–simplex → duality face ( d − 2)–simplex → k -bubble ( d − k )–simplex → ◮ Boundary triangles labeled by a color c = 0 , . . . , d 3 0 2 1 Colors identify all sub-simplices
Triangulations from colored graphs vertex d –simplex → edge ( d − 1)–simplex → duality face ( d − 2)–simplex → k -bubble ( d − k )–simplex → ◮ Boundary triangles labeled 23 by a color c = 0 , . . . , d 03 3 ◮ Induced colorings 0 2 ◮ Edges labeled by pair of 13 colors 1 02 01 12 Colors identify all sub-simplices
Triangulations from colored graphs vertex d –simplex → edge ( d − 1)–simplex → duality face ( d − 2)–simplex → k -bubble ( d − k )–simplex → 023 ◮ Boundary triangles labeled 23 by a color c = 0 , . . . , d 03 3 ◮ Induced colorings 0 2 ◮ Edges labeled by pair of 13 013 123 colors 1 02 ◮ Nodes labeled by three colors 01 12 012 Colors identify all sub-simplices
Colored attaching maps Gluing respecting all induced colorings 023 023 03 03 02 02 3 3 0 2 2 013 013 1 1 01 01 012 012 Theory of crystallization and GEMs (graph–encoded manifolds): ( d + 1)-colored graphs are dual to triangulations of pseudo-manifolds of dimension d [Pezzana, Ferri, Gagliardi, Cristofori, Casali, Lins].
The 2D case 0 0 0 0 0 0 0 2 2 0 1 1 1 1 0 0 2 2 2 2 0 1 1 0 0 0 0 0 0 0
The 2D case 0 0 0 0 0 0 0 2 2 0 1 1 1 1 0 0 2 2 2 2 0 1 1 0 0 0 0 0 0 0
The 2D case 0 0 0 0 0 0 0 2 2 0 1 1 1 1 0 0 2 2 2 2 0 1 1 0 0 0 0 0 0 0
The 2D case 0 0 0 0 0 0 0 2 2 0 1 1 1 1 0 0 2 2 2 2 0 1 1 0 0 0 0 0 0 0 2 p –angle ◮ Gluing of 2 p triangles with boundary of color 0 ◮ Dually: Components with all colors but 0
The 3D case 0 0 3 0 2 1 0 2 3 1 1 3 0 2 0 2 1 3 0 0
The 3D case 0 0 3 0 2 1 0 2 3 1 1 3 0 2 0 2 1 3 0 0
The 3D case 0 0 3 0 2 1 0 2 3 1 1 3 0 2 0 2 1 3 0 0
The 3D case 0 0 3 0 2 1 0 2 3 1 1 3 0 2 0 2 1 3 0 0
The 3D case 3 2 3 2 1 1 1 1 3 2 2 3
Bubbles as building blocks ◮ Colored graph with colors 0 , 1 , . . . , d (triangulation in dim d ) ◮ Bubble: connected piece with colors 1 , . . . , d Obtained by removing the color 0 ◮ All graphs obtained by gluing bubbles along edges of color 0 ◮ G ( B ) set of ( d + 1)-colored graphs where all bubbles are B 1 1 3 3 4 4 2 2 2 4 3 1
Bubbles II ◮ 2D: only bubbles with 2 p vertices Cycles of colors (1 , 2) 1 2
Bubbles II 1 1 2 2 2 2 3 3 3 3 1 1 1 1 ◮ 2D: only bubbles with 2 p vertices 2 2 Cycles of colors (1 , 2) 1 1 1 3 2 2 2 2 3 3 3 3 2 1 1 1 1 3 2 2 1 2 2 ◮ Many more in higher dimensions 4 4 3 1 1 1 ◮ Vast world to explore 3 3 4 3 4 2 2 3 4 1 1 2
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