## ${ℳ}_{g,n}$ counting and recursions

Last updated: 1 December 2014

## Why do I care?

$Okounkov ∣ ∣ symmetric functions = Witten Kontsevich ∣ ∣ "physics" = ℳg,n ∣ ∣ Number Theory = counting points in polytopes ∣ ∣ algebraic combinatorics ⟷ ⟷ T-equivariant cohomology ∣ ∣ Grothendieck-Lefschetz Brion-Vergne Localization of T fixed points$
$Grothendieck Dessin d'Enfants Curves over ℚ‾ ⟷ Drinfeld QuasiHopf algebras and deformations ↭ Langlands program$

(Okounkov-Pandharipande) $∫[ℳ‾g,n(X,d)]virt πc1(ℒi)ki= ∑∣λ∣=d (dim λd!)2-2genus(X) ∏i Pki+1*(X) (ki+1)!$ where $Pk*(λ)=∑i ( (λi-i+12)k- (-i+12)k ) .$ ${P}_{k}^{*}\lambda$ is the central character of a completed $k$ cycle $\frac{1}{k}{C}_{\left(k\right)}+$ smaller permutations.

## Fat graphs

A fat graph $\mathrm{\Gamma }=\left(V,E,{\text{??}}\right)$ is an undirected graph $\left(V,E\right)$ with a cyclic ordering ${Z}_{v}$ of the edges that contain $v,$ for each $v\in V,$ and such that each vertex $v\in V$ is in $>2$ edges.

A boundary component of $\mathrm{\Gamma }$ is a sequence of distinct edges $( {v0,v1},…, {vℓ-1,v0} )$ such that for $i=1,\dots ,\ell -1,$ $\left\{{v}_{i},{v}_{i+1}\right\}$ is the successor of $\left\{{v}_{0},{v}_{1}\right\}$ is the successor of $\left\{{v}_{\ell -1},{v}_{0}\right\}$ at ${v}_{0}\text{.}$ Let $n = # of boundary components of Γ, d = # of edges of Γ, d′ = # of vertices of Γ.$ The genus and Euler characteristic of $\mathrm{\Gamma }$ are $g=\frac{d\prime -d+n}{2}-1$ and $\chi =\chi \prime -d,$ respectively.

## Norbury polynomials

A metric fat graph is a pair $\left(\mathrm{\Gamma },x\right)$ where $\mathrm{\Gamma }$ is a fat graph and $x\in {ℝ}_{\ge 0}^{d}\text{.}$

The length of an edge $e$ is ${x}_{e}$ and the length of a boundary component $\left({e}_{0},\dots ,{e}_{\ell -1}\right)$ is ${x}_{{e}_{0}}+\cdots +{x}_{{e}_{\ell -1}}\text{.}$

Let $\mathrm{\Gamma }$ be a fat graph with boundary components labeled $1,2,\dots ,n\text{.}$ For ${b}_{1},\dots ,{b}_{n}\in {ℝ}_{\ge 0}$ let $PΓ(b1,…,bn) = { metric fat graphs (Γ,x) with boundary component lengths b1,…,bn } = { x∈ℝ≥0d | AΓx=b }$ where $(AΓ)eB= # of times e appears in B.$ Define $NΓ(b1,…,bn) = Card(PΓ(b1,…,bn)∩ℤr) and Ng,n(b1,…,bn) = ∑Γ∈Fatg,n NΓ(b1,…,bn) ∣Aut(Γ)∣ ,$ where $Fatg,n = { fat graphs with genus g and n boundary components labeled 1,2,…,n } , Aut(Γ) = {automorphisms of Γ}.$

## Permutations

Let $\mathrm{\Gamma }=\left(V,E,Z\right)$ be a fat graph.

Let $E‾= { (vi,vj), (vj,vi) | {vi,vj} ∈E }$ so that $\left(V,\stackrel{‾}{E}\right)$ is a directed graph with $2d$ edges.

Define ${\sigma }_{0},{\sigma }_{\frac{1}{2}}\in {S}_{2d}$ by $σ12((vi,vj)) =(vj,vi),$ and $(σ0σ12) ((vi,vj))= (vj,vk),$ where $\left\{{v}_{j},{v}_{k}\right\}$ is the successor of $\left\{{v}_{i},{v}_{j}\right\}$ at ${v}_{j}\text{.}$ Then, in cycle notation $σ∞-1=σ0 σ12=(B1,…,Bn),$ where ${B}_{i}$ are the boundary components of $\mathrm{\Gamma },$ and $σ0σ12σ∞=1,$ in ${S}_{d}\text{.}$

$Γ= {e}_{1} {e}_{2} {e}_{3} , Z1 = (e1e2e3), Z2 = (e1e3e2),$ $B1 = (e1e2), B2 = (e2e3), B3 = (e1e3) ,AΓ= ( 110 011 101 ) ,$ ${e}_{1} {e}_{2} {e}_{3} {\stackrel{‾}{e}}_{1} {\stackrel{‾}{e}}_{2} {\stackrel{‾}{e}}_{3} σ0 = 1 \stackrel{‾}{1} 2 \stackrel{‾}{2} 3 \stackrel{‾}{3} , σ12 = 1 \stackrel{‾}{1} 2 \stackrel{‾}{2} 3 \stackrel{‾}{3} .$ Then ${\sigma }_{\infty }^{-1}={\sigma }_{0}{\sigma }_{\frac{1}{2}}$ has cycles $\left({e}_{1}{e}_{2}\right)\left({\stackrel{‾}{e}}_{1}{\stackrel{‾}{e}}_{3}\right)\left({\stackrel{‾}{e}}_{2}{\stackrel{‾}{e}}_{3}\right)$ and $σ0σ12σ∞=1.$ To consider the metric fat graph $\left(\mathrm{\Gamma },\left(2,2,1\right)\right)$ insert vertices in ${e}_{1}$ and ${e}_{2}\text{.}$ ${e}_{1} {e}_{2} {e}_{3} {e}_{4} {e}_{5} {\stackrel{‾}{e}}_{1} {\stackrel{‾}{e}}_{2} {\stackrel{‾}{e}}_{3} {\stackrel{‾}{e}}_{4} {\stackrel{‾}{e}}_{5}$ Then $σ0 = 1 \stackrel{‾}{1} 2 \stackrel{‾}{2} 3 \stackrel{‾}{3} 4 \stackrel{‾}{4} 5 \stackrel{‾}{5} , σ12 = 1 \stackrel{‾}{1} 2 \stackrel{‾}{2} 3 \stackrel{‾}{3} 4 \stackrel{‾}{4} 5 \stackrel{‾}{5}$ has cycles $(e1,e2,e4,e5), (e‾1,e‾5,e‾3), (e‾2,e3,e‾4)$ and $\left({b}_{1},{b}_{2},{b}_{3}\right)=\left(4,3,3\right)\text{.}$

## Base case examples

$\left(g,n\right)=\left(0,3\right)$ where ${N}_{0,3}=1\text{.}$ There are $7$ labeled fat graphs and $3$ unlabeled fat graphs. $Γ = {e}_{1} {e}_{2} {e}_{3} , AΓ = ( 211 010 001 ) , Γ = {e}_{1} {e}_{2} {e}_{3} , AΓ = ( 110 011 101 ) , Γ = {e}_{1} {e}_{2} , AΓ = ( 11 10 01 ) .$

$\left(g,n\right)=\left(1,1\right)$ where ${N}_{1,1}\left({b}_{1}\right)=\frac{1}{48}\left({b}_{1}^{2}-4\right)\text{.}$ $Γ = {e}_{1} {e}_{2} {e}_{3} , AΓ = (2 2 2), ∣Aut Γ∣ = 6, Γ = {e}_{1} {e}_{2} , AΓ = (2 2), σ0 = 1 \stackrel{‾}{1} 2 \stackrel{‾}{2} σ12 = 1 \stackrel{‾}{1} 2 \stackrel{‾}{2} Aut Γ = ℤ/4ℤ.$

## Relation to ${ℳ}_{g,n}$

$ℳg,n= {genus g curves with n marked points},$ $curve = Riemann surface,oriented,conformal class… connected,compact…$ $ℳg,ncomb (b1,…,bn)= ⋃Γ∈Fatg,n PΓ(b1,…,bn) ≃ℳg,n.$

## Relation to branched covers of ${ℙ}^{1}$

A branched cover of ${ℙ}^{1}$ is a map $Σ ↓ ℙ1$ where $\mathrm{\Sigma }$ is ....

Three permutations ${\sigma }_{0},{\sigma }_{\frac{1}{2}},{\sigma }_{\infty }\in {S}_{2d}$ such that $\mathrm{\Sigma } {ℙ}^{1} {\sigma }_{0} {\sigma }_{\frac{1}{2}} {\sigma }_{\infty } 0 \frac{1}{2} \infty$ ${\sigma }_{0}{\sigma }_{\frac{1}{2}}{\sigma }_{\infty }=1$ specify a degree $d$ branched cover of ${ℙ}^{1}\text{.}$

## Notes and References

This is a typed copy of notes for the Algebra, Geometry and Topology seminar given on August 4, 2008 at the University of Melbourne.