[PPT] - Maps A map is an equivalence class of labeled graphs embedded on a PowerPoint Presentation

SLIDE 1

Maps

A map is an equivalence class of labeled graphs embedded on a compact Riemann surface. Equivalent– if an orientation preserving homeomorphism of the surface takes one graph to the other. Map condition– the graph’s complement must be a disjoint union of topological discs (faces).

SLIDE 2

Maps

A map is an equivalence class of labeled graphs embedded on a compact Riemann surface. Equivalent– if an orientation preserving homeomorphism of the surface takes one graph to the other. Map condition– the graph’s complement must be a disjoint union of topological discs (faces). Labels–

The vertices have distinct names We choose a function assigning to each vertex one of its incident edges

SLIDE 3

Edges cannot intersect

1

No Yes

1 2

SLIDE 4

Faces must be discs

1

Yes

SLIDE 5

Faces must be discs

1

Yes

1

No

SLIDE 6

Dehn twist

1

These are the same map

1

SLIDE 7

How can log Zn know about maps???

SLIDE 8

GUE covariances

Gaussian Unitary Ensemble: dPn(M) = 1 Zn e− 1

2 Tr M2 dM

E[MijMkl] =δi=lδj=k

SLIDE 9

Wick’s lemma

f1, . . . , f2m are linear functionals on Rn×n. E[f1 . . . f2m] =

w∈Wick pairings
f 1, . . . , 2m

E[fw(1)fw(2)] . . . E[fw(2m−1)fw(2m)] An example of a Wick pairing of 1, . . . , 8 is {{1, 6}, {2, 5}, {3, 4}, {7, 8}}.

SLIDE 10

Matrix Integrals and combinatorics

Here is a very brief hand wave at the connection between matrix integrals in map combinatorics. E

Tr M4p

=E  

p

q=1

n

iq,jq,kq,lq=1

Miq,jqMjq,kqMkq,lqMlq,iq   =

n

i1,j1,k1,...

,jp,kp,lp=1

w∈Wick pairings
f 1,. . . ,4p
Product of quadratic

expectations given by w and index variables

=
4-valent fatgraphs
n p vertices

nFaces

SLIDE 11

A Wick pairing and the corresponding map

The Wick pairing (i11, j13), (i13, k23), (k13, k33), (i23, i33), (j23, j33) corresponds to a fatgraph and a map.

1 2

3

j k

i1 i i

j j k k

1 1 2 2 2 3 3 3

3

3 3 3

3 3 3 3 3 3 3 3

1

i1

1

SLIDE 12