${ℳ}_{g,n}$ counting and recursions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 1 December 2014

Why do I care?

$$\begin{array}{ccccccc}\begin{array}{c}\text{Okounkov}\\ \mid \hspace{0.17em}\mid \\ \text{symmetric functions}\end{array}& =& \begin{array}{c}\text{Witten Kontsevich}\\ \mid \hspace{0.17em}\mid \\ \text{"physics"}\end{array}& =& \begin{array}{c}{ℳ}_{g,n}\\ \mid \hspace{0.17em}\mid \\ \text{Number Theory}\end{array}& =& \begin{array}{c}\text{counting points in polytopes}\\ \mid \hspace{0.17em}\mid \\ \text{algebraic combinatorics}\end{array}\\ & \begin{array}{c}⟷\end{array}& \multicolumn{3}{c}{}& \begin{array}{c}⟷\end{array}\\ \multicolumn{2}{c}{}& \multicolumn{3}{c}{T\text{-equivariant cohomology}}\\ \multicolumn{2}{c}{}& \multicolumn{3}{c}{\mid \hspace{0.17em}\mid }\\ \multicolumn{2}{c}{}& \multicolumn{3}{c}{\begin{array}{c}\text{Grothendieck-Lefschetz}\\ \text{Brion-Vergne}\\ \text{Localization of}\hspace{0.17em}T\hspace{0.17em}\text{fixed points}\end{array}}\end{array}$$
$$\begin{array}{ccccc}\begin{array}{c}\text{Grothendieck}\\ \text{Dessin d'Enfants}\\ \text{Curves over}\hspace{0.17em}\bar{\mathbb{Q}}\end{array}& ⟷& \begin{array}{c}\text{Drinfeld}\\ \text{QuasiHopf}\\ \text{algebras and}\\ \text{deformations}\end{array}& \leftrightsquigarrow & \begin{array}{c}\text{Langlands}\\ \text{program}\end{array}\end{array}$$

(Okounkov-Pandharipande) $$\underset{{\left[{\overline{ℳ}}_{g,n}(X,d)\right]}^{\text{virt}}}{\int }\pi c_1{\left({ℒ}_i\right)}^{k_i}=\sum _{\mid \lambda \mid =d}{\left(\frac{\text{dim}\hspace{0.17em}\lambda }{d!}\right)}^{2-2\text{genus}\left(X\right)}\prod _i\frac{P_{k_i+1}^{*}\left(X\right)}{(k_i+1)!}$$ where $$P_k^{*}\left(\lambda \right)=\sum _i({({\lambda }_i-i+\frac{1}{2})}^k-{(-i+\frac{1}{2})}^k)\text{.}$$ $P_k^{*}\lambda$ is the central character of a completed $k$ cycle $\frac{1}{k}{C}_{\left(k\right)}+$ smaller permutations.

Fat graphs

A fat graph $\mathrm{\Gamma }=(V,E,\text{??})$ is an undirected graph $(V,E)$ 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 $$(\{v_0,v_1\},\dots ,\{v_{\ell -1},v_0\})$$ such that for $i=1,\dots ,\ell -1,$ $\{v_i,v_{i+1}\}$ is the successor of $\{v_0,v_1\}$ is the successor of $\{v_{\ell -1},v_0\}$ at $v_0\text{.}$ Let $$\begin{array}{rcl}{\displaystyle n}& =& {\displaystyle \text{\# of boundary components of}\hspace{0.17em}\mathrm{\Gamma },}\\ {\displaystyle d}& =& {\displaystyle \text{\# of edges of}\hspace{0.17em}\mathrm{\Gamma },}\\ {\displaystyle d\prime }& =& {\displaystyle \text{\# of vertices of}\hspace{0.17em}\mathrm{\Gamma }\text{.}}\end{array}$$ 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 $(\mathrm{\Gamma },x)$ 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 $(e_0,\dots ,e_{\ell -1})$ 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 $$\begin{array}{rcl}{\displaystyle P_{\mathrm{\Gamma }}(b_1,\dots ,b_n)}& =& {\displaystyle \left\{\begin{array}{c}\text{metric fat graphs}\hspace{0.17em}(\mathrm{\Gamma },x)\hspace{0.17em}\text{with}\\ \text{boundary component lengths}\hspace{0.17em}{b}_1,\dots ,b_n\end{array}\right\}}\\ & =& {\displaystyle \{x\in {ℝ}_{\ge 0}^d\hspace{0.17em}|\hspace{0.17em}{A}_{\mathrm{\Gamma }}x=b\}}\end{array}$$ where $${\left(A_{\mathrm{\Gamma }}\right)}_{eB}=\text{\# of times}\hspace{0.17em}e\hspace{0.17em}\text{appears in}\hspace{0.17em}B\text{.}$$ Define $$\begin{array}{rcl}{\displaystyle N_{\mathrm{\Gamma }}(b_1,\dots ,b_n)}& =& {\displaystyle \text{Card}(P_{\mathrm{\Gamma }}(b_1,\dots ,b_n)\cap {\mathbb{Z}}^r)\text{and}}\\ {\displaystyle N_{g,n}(b_1,\dots ,b_n)}& =& {\displaystyle \sum _{\mathrm{\Gamma }\in {\text{Fat}}_{g,n}}\frac{N_{\mathrm{\Gamma }}(b_1,\dots ,b_n)}{\mid \text{Aut}\left(\mathrm{\Gamma }\right)\mid },}\end{array}$$ where $$\begin{array}{rcl}{\displaystyle {\text{Fat}}_{g,n}}& =& {\displaystyle \{\text{fat graphs with genus}\hspace{0.17em}g\hspace{0.17em}\text{and}\hspace{0.17em}n\hspace{0.17em}\text{boundary components labeled}\hspace{0.17em}1,2,\dots ,n\},}\\ {\displaystyle \text{Aut}\left(\mathrm{\Gamma }\right)}& =& {\displaystyle \left\{\text{automorphisms of}\hspace{0.17em}\mathrm{\Gamma }\right\}\text{.}}\end{array}$$

Permutations

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

Let $$\bar{E}=\{(v_i,v_j),(v_j,v_i)\hspace{0.17em}|\hspace{0.17em}\{v_i,v_j\}\in E\}$$ so that $(V,\bar{E})$ is a directed graph with $2d$ edges.

Define ${\sigma }_0,{\sigma }_{\frac{1}{2}}\in S_{2d}$ by $${\sigma }_{\frac{1}{2}}\left((v_i,v_j)\right)=(v_j,v_i),$$ and $$\left({\sigma }_0{\sigma }_{\frac{1}{2}}\right)\left((v_i,v_j)\right)=(v_j,v_k),$$ where $\{v_j,v_k\}$ is the successor of $\{v_i,v_j\}$ at $v_j\text{.}$ Then, in cycle notation $${\sigma }_{\infty }^{-1}={\sigma }_0{\sigma }_{\frac{1}{2}}=(B_1,\dots ,B_n),$$ where $B_i$ are the boundary components of $\mathrm{\Gamma },$ and $${\sigma }_0{\sigma }_{\frac{1}{2}}{\sigma }_{\infty }=1,$$ in $S_d\text{.}$

$$\mathrm{\Gamma }=\begin{array}{c} e1 e2 e3 \end{array},\begin{array}{rcl}{\displaystyle Z_1}& =& {\displaystyle \left(e_1{e}_2{e}_3\right),}\\ {\displaystyle Z_2}& =& {\displaystyle \left(e_1{e}_3{e}_2\right),}\end{array}$$ $$\begin{array}{rcl}{\displaystyle B_1}& =& {\displaystyle \left(e_1{e}_2\right),}\\ {\displaystyle B_2}& =& {\displaystyle \left(e_2{e}_3\right),}\\ {\displaystyle B_3}& =& {\displaystyle \left(e_1{e}_3\right)}\end{array},A_{\mathrm{\Gamma }}=\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 1& 0& 1\end{array}\right),$$ $$\begin{array}{c} e1 e2 e3 e‾1 e‾2 e‾3 \end{array}\begin{array}{rcl}{\displaystyle {\sigma }_0}& =& {\displaystyle \begin{array}{c} 1 1‾ 2 2‾ 3 3‾ \end{array},}\\ {\displaystyle {\sigma }_{\frac{1}{2}}}& =& {\displaystyle \begin{array}{c} 1 1‾ 2 2‾ 3 3‾ \end{array}\text{.}}\end{array}$$ Then ${\sigma }_{\infty }^{-1}={\sigma }_0{\sigma }_{\frac{1}{2}}$ has cycles $\left(e_1{e}_2\right)\left({\bar{e}}_1{\bar{e}}_3\right)\left({\bar{e}}_2{\bar{e}}_3\right)$ and $${\sigma }_0{\sigma }_{\frac{1}{2}}{\sigma }_{\infty }=1\text{.}$$ To consider the metric fat graph $(\mathrm{\Gamma },(2,2,1))$ insert vertices in $e_1$ and $e_2\text{.}$ $$\begin{array}{c} e1 e2 e3 e4 e5 e‾1 e‾2 e‾3 e‾4 e‾5 \end{array}$$ Then $$\begin{array}{rcl}{\displaystyle {\sigma }_0}& =& {\displaystyle \begin{array}{c} 1 1‾ 2 2‾ 3 3‾ 4 4‾ 5 5‾ \end{array},}\\ {\displaystyle {\sigma }_{\frac{1}{2}}}& =& {\displaystyle \begin{array}{c} 1 1‾ 2 2‾ 3 3‾ 4 4‾ 5 5‾ \end{array}}\end{array}$$ has cycles $$(e_1,e_2,e_4,e_5),({\bar{e}}_1,{\bar{e}}_5,{\bar{e}}_3),({\bar{e}}_2,e_3,{\bar{e}}_4)$$ and $(b_1,b_2,b_3)=(4,3,3)\text{.}$

Base case examples

$(g,n)=(0,3)$ where $N_{0,3}=1\text{.}$ There are $7$ labeled fat graphs and $3$ unlabeled fat graphs. $$\begin{array}{rclllll}{\displaystyle \mathrm{\Gamma }}& =& {\displaystyle \begin{array}{c} e1 e2 e3 \end{array},}& & {\displaystyle A_{\mathrm{\Gamma }}}& =& {\displaystyle \left(\begin{array}{ccc}2& 1& 1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),}\\ {\displaystyle \mathrm{\Gamma }}& =& {\displaystyle \begin{array}{c} e1 e2 e3 \end{array},}& & {\displaystyle A_{\mathrm{\Gamma }}}& =& {\displaystyle \left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 1& 0& 1\end{array}\right),}\\ {\displaystyle \mathrm{\Gamma }}& =& {\displaystyle \begin{array}{c} e1 e2 \end{array},}& & {\displaystyle A_{\mathrm{\Gamma }}}& =& {\displaystyle \left(\begin{array}{cc}1& 1\\ 1& 0\\ 0& 1\end{array}\right)\text{.}}\end{array}$$

$(g,n)=(1,1)$ where $N_{1,1}\left(b_1\right)=\frac{1}{48}(b_1^2-4)\text{.}$ $$\begin{array}{rclllllllll}{\displaystyle \mathrm{\Gamma }}& =& {\displaystyle \begin{array}{c} e1 e2 e3 \end{array},}& & {\displaystyle A_{\mathrm{\Gamma }}}& =& {\displaystyle \left(2\hspace{0.17em}2\hspace{0.17em}2\right),}& & {\displaystyle \mid \text{Aut}\hspace{0.17em}\mathrm{\Gamma }\mid }& =& {\displaystyle 6,}\\ {\displaystyle \mathrm{\Gamma }}& =& {\displaystyle \begin{array}{c} e1 e2 \end{array},}& & {\displaystyle A_{\mathrm{\Gamma }}}& =& {\displaystyle \left(2\hspace{0.17em}2\right),\begin{array}{rcl}{\displaystyle {\sigma }_0}& =& {\displaystyle \begin{array}{c} 1 1‾ 2 2‾ \end{array}}\\ {\displaystyle {\sigma }_{\frac{1}{2}}}& =& {\displaystyle \begin{array}{c} 1 1‾ 2 2‾ \end{array}}\end{array}}& & {\displaystyle \text{Aut}\hspace{0.17em}\mathrm{\Gamma }}& =& {\displaystyle \mathbb{Z}/4\mathbb{Z}\text{.}}\end{array}$$

Relation to ${ℳ}_{g,n}$

$${ℳ}_{g,n}=\left\{\text{genus}\hspace{0.17em}g\hspace{0.17em}\text{curves with}\hspace{0.17em}n\hspace{0.17em}\text{marked points}\right\},$$ $$\begin{array}{rcl}{\displaystyle \text{curve}}& =& {\displaystyle \begin{array}{c}\text{Riemann surface,}\text{oriented,}\text{conformal class}\dots \\ \text{connected,}\text{compact}\dots \end{array}}\end{array}$$ $${ℳ}_{g,n}^{\text{comb}}(b_1,\dots ,b_n)=\bigcup _{\mathrm{\Gamma }\in {\text{Fat}}_{g,n}}{P}_{\mathrm{\Gamma }}(b_1,\dots ,b_n)\simeq {ℳ}_{g,n}\text{.}$$

Relation to branched covers of ${\mathbb{P}}^1$

A branched cover of ${\mathbb{P}}^1$ is a map $$\begin{array}{c}\mathrm{\Sigma }\\ \downarrow \\ {\mathbb{P}}^1\end{array}$$ where $\mathrm{\Sigma }$ is ....

Three permutations ${\sigma }_0,{\sigma }_{\frac{1}{2}},{\sigma }_{\infty }\in S_{2d}$ such that $$\begin{array}{c} Σ ℙ1 σ0 σ12 σ∞ 0 12 ∞ \end{array}$$ ${\sigma }_0{\sigma }_{\frac{1}{2}}{\sigma }_{\infty }=1$ specify a degree $d$ branched cover of ${\mathbb{P}}^1\text{.}$

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.

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